3.4 Stochastic Models
3.4.3 Three Factor Models
showing the last equation of Table 3.1.
2 [Schwartz Smith 2000] showed that the model works best for mid term maturities. Moreover, the model includes the two one factor models Brownian Motion and Mean Reversion. The first one is generated by setting σχ equal to zero, i.e. assuming that
there is uncertainty in equilibrium prices, only. A Mean Reversion Model is given by assuming a constant equilibrium price, i.e. setting σξ equal to zero. Statistical
comparison of the three models by the authors showed significant advantages in cap- turing the characteristics of commodity futures prices through the two factor model. But as [Lautier 2005] stats, there is still one question remaining: is it interesting to represent a stable equilibrium with a stochastic variable? On the other hand, some pricing perspectives, especially in the real options environment, focus on long term prices and do not care about short term fluctuation.82
3.4.3 Three Factor Models
Not until 1997, the first three factor model was introduced: [Schwartz 1997] pro- posed his three factor model with the extension of stochastic interest rates because the hypothesis of constant interest rates as in the one and two factor models amounts to saying that the term structure of interest rates is flat, which is far from reality. Moreover, under this assumption forward and futures prices are equivalent, which is not the case.83 With a stochastic interest rate, it is possible to determine two
distinct payoff structures for forwards and futures, i.e. to take into account the margin call mechanism of the futures market. Finally, following [Lautier 2005], the
82Compare [Schwartz 1998].
83See [Pindyck 1994] and [French 1983]. Compare Section 2.3.1.1 Paragraph ”Forwards and Fu- tures”.
presence of the interest rate as a third explicative factor is consistent with the theory of storage. When interest rates are high, storage is more expensive resulting into a reduction of inventory and therewith, increasing the convenience yield.
Definition 3.5 Convenience Yield Model
Let P (t) be the spot price of a commodity at time t ∈ [0, T ], c(t) the convenience yield at time t, r(t) the interest rate at time t, µ ∈ R the drift of the spot price, α ∈ R is the long-run level to which the convenience yield reverts, m ∈ R is the long-run level to which the interest rate reverts, κ > 0 is the speed of adjustment of the convenience yield, β > 0 is the speed of adjustment of the interest rate, σP > 0 the spot price volatility, σc > 0 the convenience yield volatility, σr > 0
the interest rate volatility and dWP(t), dWc(t) and dWr(t) the increments of three
Brownian Motions as defined in Definition C.28 with the following correlations: dWP(t)dWc(t) = ρP cdt, dWc(t)dWr(t) = ρcrdt and dWr(t)dWP(t) = ρrPdt, with
t ∈ [0, T ] and ρ ∈ [−1, 1]. The spot price, the instantaneous convenience yield and the the instantaneous interest rate process are assumed to have the following form:
dP (t) = P (t)[µ − c(t)]dt + σPP (t)dWP(t), (3.45)
dc(t) = κ[α − c(t)]dt + σcdWc(t), (3.46)
dr(t) = β[m − r(t)]dt + σrdWr(t), t ∈ [0, T ] (3.47)
The stochastic factors in the models are the commodity spot price, the convenience yield and the interest rate. By assuming a simple mean reverting process for the interest rate, it is possible to obtain a closed form solution for futures prices. Their derivation can be found in [Schwartz 1997].
A new approach comes from [Cortazar Schwartz 2003] as introduced in Definition 3.6. Again, the spot price and the convenience yield are the first two risk factors but as third they consider the long term spot price return, allowing it to be stochastic and to return to a long term average. The temporary price variations are assumed to be activated by changes in inventory, whereas the long term return is due to changes in technologies, inflation or demand pattern. The dynamics are modeled as follows:
Definition 3.6 The Long Term - Convenience Yield Model
Let P (t) be the spot price of a commodity at time t ∈ [0, T ], y(t) the demeaned convenience yield at time t, with y := c − α, where α is the long run mean of the convenience yield c, v(t) the expected long-term spot price return at time t, with v := µ−α, where µ ∈ R is the drift of P , κ, a > 0 the speed of adjustments of the demeaned convenience yield of v; ¯v ∈ R the long-run mean of the expected long-term spot price
return, σP, σy, σv > 0 the corresponding volatilities and dW (t)P, dW (t)y, dW (t)v the
increments of three standard Brownian Motions as defined in Definition C.28 with correlations ρP y, ρP v, ρyv ∈ [−1, 1]. Then the dynamic of this model is
dP (t) = P (t)[v(t) − y(t)]dt + σPP (t)dWP(t), (3.48)
dy(t) = −κy(t)dt + σydWy(t), (3.49)
dv(t) = a[¯v − v(t)]dt + σvdWv(t), t ∈ [0, T ]. (3.50)
In practice, the development of three factor models deposit the question of sense and usage. Although an empirical comparison of three factor to two factor models show that the introduction of a third factor improves the performance of the models in terms of their ability to describe the evolution of futures prices, this improvement is too small to justify for higher computational costs. Especially, for the evaluation of more complex derivatives parsimony is needed. [Schwartz 1997] concludes, that the two factor Convenience Yield Model has the best return on investment.
In the following section we will introduce the commodity trading vehicle our main focus is put on over the next sections: commodity indices. In traditional financial markets, indices are assumed to produce attractive risk and return profiles. But what about commodity markets? Indeed, commodity indices represent diversified portfolios participating from the different facilities of their elements.84 Recall, a
commodity investment over a CTA also provides diversified commodity exposure. But the main difference between commodity indices and managed futures accounts or funds is, that the indices introduced in this section represent long only, buy and hold strategies whereby CTAs actively trade commodity derivatives, i.e. they are allowed to trade short positions for instance. This yields to different risk and return structures. [Schneeweiss Spurgin 1996] analyzed various commodity indices and indices which are used to track managed futures performance. Results indicate that a buy and hold commodity investment strategy provides a poor forecast of CTA returns. Therefore, commodity indices have to be treated differently.
Because commodity investment is still adolescent, there is only a very little amount of commodity indices of less than 20 available. They differ among each other by e.g. number of commodities involved, their weighting and rebalancing procedures. The different characteristics are described in Section 4.1 and shall serve us as a first warming up. The following Section 4.2 provides information about the major commodity indices. Most of them are not older than ten to 15 years and there are partly huge creation differences among them.
Investors are accustomed and attracted to the ability of entering a market via cheap diversified exposure yielding into an increasing demand for commodity linked prod- ucts that will be introduced in Section 4.3. Products like mutual or exchange traded funds tracking an index are known from stock and bond markets and famous. Es- pecially the fees are much lower than managed futures fund fees that can yield up to 25%.
We already know from Section 3 the source of commodity futures return evolving in changes of the current supply and demand equilibriums. We will close this section in 4.4 by decomposing commodity index returns and filtering their origins.
4.1 Characteristics
A commodity index is designed to represent and track price changes in a basket of commodity futures contracts. The concept of an investable commodity index that is treated as a separate asset class was first introduced in [Greer 1978]. The underlying logic is that the returns on the index approximate the returns to an investor holding a position in the assets underlying basket. Following [Structured Products 2006] the difference between the different indexes is based on a variety of design factors: